The Prime Minister said the Government took "prompt and decisive" action when the mutant variant was discovered in the UK on Friday, resulting in the creation of strict Tier 4 restrictions that have taken a scythe to Christmas plans for millions.
He said more than half a million people in the UK have now received their first dose of the two-part vaccination since British grandmother Margaret Keenan became the first patient in the world on December 8 to receive the Pfizer Covid-19 jab following its clinical approval.
It came as the Government's New and Emerging Respiratory Virus Threats advisory group (NervTag) told a Science Media Centre briefing on the new mutant variant that cases outside of Tier 4 in London and the south east of England "are increasing at similar rates" those in affected areas, and that it had a "transmission advantage" over other strains of the virus.
However, World Health Organisation director-general Dr Tedros Adhanom Ghebreyesus said there was "no evidence" the new strain was "more likely to cause severe disease or mortality".
Elsewhere, regional public health directors in Manchester and the West Midlands urged anyone who travelled from a Tier 4 area or Wales to self isolate upon their arrival and "assume" they have the new Covid-19 variant.
Speaking at a Downing Street press conference on Monday evening, after crunch talks involving members of the Cobra emergency committee to prevent the UK being cut off from the continent in the days before Christmas, Mr Johnson said discussions were taking place to "unblock the flow of trade as fast as possible".
He said: "It was an excellent conversation with the French President (Emmanuel Macron), he stressed he was keen, I would say, to sort it out in the next few hours if we can.
"Our teams will be working on it flat out - if we can get a result then, that would be great, but we will do it as fast as we can."
Transport Secretary Grant Shapps urged people not to travel to Kent amid the closure of the French border.
He told the press conference: "Please don't travel to Kent.
"Most people should be staying at home, everybody in Tier 4 must at stay home and in Tier 3 stay very local."
Referring to the vaccine, Mr Johnson said: "I can today announce that over half a million people, more than 500,000 people in the UK, have now received their first dose."
Mr Johnson said the "vast bulk" of the initial 500,000 doses had gone to people aged over 80 and those who were clinically vulnerable.
But Sir Patrick Vallance, the Government's chief scientific adviser, warned the new variant was "spread around the country", and urged people to "stay local".
He told the Downing Street press conference: "People shouldn't be travelling around the country at the moment."
The closure of cross-Channel routes overnight alarmed businesses, including those relying on the trouble-free passage of produce into the UK, as well as holidaymakers looking to leave for the continent - all with the added complication of the end of the Brexit transition period on December 31.
Supermarket giant Sainsbury's warned that disruption in Kent, where more than 170 lorries remain backed up on roads around the Port of Dover, could hit supplies of lettuce, some salad leaves, cauliflowers, broccoli and citrus fruit, all of which are imported from mainland Europe in the winter.
But French transport minister Jean-Baptiste Djebarri said he hoped a protocol would be agreed by European Union states "to ensure that movement from the UK can resume" after his country banned passengers and freight crossing the English Channel.
Measures being considered in France include the possibility of testing the country's nationals to allow them to return from the UK.
Transport Secretary Mr Shapps said emergency measures were being put in place to cope with a backlog of lorries heading for the Channel ports.
But he sought to play down the potential impact, stressing that container freight was not hit by the ban on people crossing the border and hauliers were "quite used to anticipating disruption".
The disused Manston Airport in Kent would be used as a lorry park, while Operation Stack, the contingency measures used to queue on the M20 whenever there is disruption at the Channel, was already in place.
It meant hauliers from across Europe were faced with spending the night in their cabs.
The Transport Secretary dismissed concerns about the impact of the travel ban on supplies of the Pfizer/BioNTech vaccine, which is manufactured in Belgium, because container freight was unaffected.
Port of Dover chief executive Doug Bannister said that a prolonged period of disruption would be a "stark situation".
"Because of the importance of the Dover Strait in handling critical goods such as food and other things like that I think it could become quite dramatic," he warned.
It came as the Government said a further 215 people had died within 28 days of testing positive for Covid-19 as of Monday, bringing the UK total to 67,616.
It said that, as of 9am on Sunday, there had been a further 33,364 lab-confirmed cases of coronavirus in the UK.
It brings the total number of cases in the UK to 2,073,511.